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Unformatted text preview: 1 Lecture XIX Visualizing Vector Fields; Line Integrals Visualizing Vector Fields Recall that a vector field in E 2 is a function of the form F ( x, y ) = f 1 ( x, y ) ˆ i + f 2 ( x, y ) ˆ j. We define two concepts that help us visualize vector fields. Definition 1 Let C be a directed smooth curve in E 2 , and let F be a vector field in E 2 . Then C is called an integral curve of F if, at any point P on C , F ( P ) = 0 and F ( P ) has the same direction F ( P ) is tangent to the curve C , i.e. as T ˆ P , the unit tangent vector to C at P . Recall that C is the class of all continuous scalar functions and that C 1 is the class of all differentiable functions with continuous partial derivatives. We say that the vector field F as defined above is in C 1 if f 1 , f 2 are in C 1 . Lemma 1 For any point P such that F ( P ) = 0 , there exists an integral curve for F through P . F ˆ Let us take = x ˆ i + yj = rr ˆ. The integral curves of F are rays comming out of 1 the origin. F passes the derivative test, and it is easy to see that...
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